A2 October 2021 Q7

EdexcelCurrent spec12 marksGame Theory

7. Alexis and Becky are playing a zero-sum game.

Alexis has two options, Q and R. Becky has three options, X, Y and Z.

Alexis intends to make a random choice between options Q and R, choosing option Q with probability \(p_1\) and option R with probability \(p_2\)

Alexis wants to find the optimal values of \(p_1\) and \(p_2\) and formulates the following linear programme, writing the constraints as inequalities.

Maximise \(P = V\)

where \(V = 3 +\) the value of the game to Alexis

\[\begin{array}{ll} \text{subject to} & V \leqslant 6p_1 + p_2 \\ & V \leqslant 8p_2 \\ & V \leqslant 4p_1 + 2p_2 \\ & p_1 + p_2 \leqslant 1 \\ & p_1 \geqslant 0,\ p_2 \geqslant 0,\ V \geqslant 0 \end{array}\]

(a) Complete the pay-off matrix for Alexis below. (2)

	Option X	Option Y	Option Z
Option Q
Option R

(b) Use a graphical method to find the best strategy for Alexis. (6)

Becky intends to make a random choice between options X, Y and Z, choosing option X with probability \(q_1\), option Y with probability \(q_2\) and option Z with probability \(q_3\)

(d) Determine the best strategy for Becky, making your method and working clear. (3)

Mark scheme (a)

Scheme

Marks

	Option X	Option Y	Option Z
Option Q	\(3\)	\(-3\)	\(1\)
Option R	\(-2\)	\(5\)	\(-1\)

M1
A1

3.1a
2.2a

(2)

Notes

M1: Either one correct row or column

A1: cao

SC M1 A0 for \(\begin{pmatrix} 6 & 0 & 4 \\ 1 & 8 & 2 \end{pmatrix}\)

Graph with vertical axes at p1 = 0 and p1 = 1 marked 0 to 8: line from 1 to 6 (5p1 + 1), line from 8 to 0 (−8p1 + 8) and line from 2 to 4 (2p1 + 2) — Mark scheme (b)

Scheme	Marks	AO
If \(B\) plays option X, \(A\)’s gains are \(6p_1 + p_2 = 6p_1 + (1 - p_1) = 5p_1 + 1\) If \(B\) plays option Y, \(A\)’s gains are \(8p_2 = 8(1 - p_1) = -8p_1 + 8\) If \(B\) plays option Z, \(A\)’s gains are \(4p_1 + 2p_2 = 4p_1 + 2(1 - p_1) = 2p_1 + 2\)	M1 A1	3.1a 1.1b
	M1 A1	1.1b 1.1b
\(2 + 2p_1 = 8 - 8p_1 \Rightarrow p_1 = 0.6\)	A1	1.1b
Alexis should play option Q with probability 0.6 and option R with probability 0.4	A1ft	3.2a
	(6)

Scheme	Marks	AO
Value of the game \(= 2 + 2\left(\dfrac{3}{5}\right) - 3 = \dfrac{1}{5}\)	B1	2.2a
	(1)

Mark scheme (d)

Scheme	Marks	AO
\(4q_3 = \dfrac{16}{5},\ 8q_2 + 2q_3 = \dfrac{16}{5}\) or \(-3q_2 + q_3 = \dfrac{1}{5},\ 5q_2 - q_3 = \dfrac{1}{5}\)	M1 A1	3.1a 1.1b
\(q_2 = \dfrac{1}{5},\ q_3 = \dfrac{4}{5} \Rightarrow\) Becky should play option X never, option Y with probability 0.2 and option Z with probability 0.8	A1	3.2a
	(3)
	(12 marks)

Notes

M1: Setting up two equations in \(q_2\) and \(q_3\) with their value of either the original or modified game

A1: Correct two equations

A1: Interpret their values in the context of the question – must refer to play and the associated probabilities

For part (d) candidates might set up three equations in three unknowns e.g.,

\[\begin{aligned} 3q_1 - 3q_2 + q_3 &= \frac{1}{5} \\ -2q_1 + 5q_2 - q_3 &= \frac{1}{5} \\ q_1 + q_2 + q_3 &= 1 \end{aligned}\]